Aircraft use the traditional XYZ coordinate system, but it is oriented a little bit differently than you may expect. On a plane, the positive x direction is pointing in the direction of the nose, the positive y direction is pointing along the right wing, and the positive z direction is pointing downwards towards the ground. An image of this can be found in Figure 1.
Fig. 1 - Airplane Axes
In Figure 1, there are a few things going on. For starters, the x, y, and z directions are labeled and colored in red, green, and purple, respectively. Now notice each axis has a curved arrow going around it; this is an indication of the positive direction of rotation about each axis. Another way you can remember the direction of the positive rotation is via right-hand rule. Stick your right thumb in the direction of the axis and curl your fingers, that is the positive direction of rotation!
There is also notation for movement in the xyz directions (velocity). This is often noted as u (x-dir), v (y-dir), and w (z-dir) for the velocities in each respective direction. In addition to translational velocity, there is a notation for rotational velocity as well. Rotational velocities are noted as p (x-dir), q (y-dir), and r (z-dir). Finally, there is also a specific notation for moments, or torques, about an axis. They are noted as L (x-dir), M (y-dir), and N (z-dir). Rotation about the x-axes is called Roll, rotation about the y-axes is called Pitch, and rotation about the z-axes is called Yaw.
With an aircraft, it is common for Euler angles to be used for describing its current position. There are 3 different Euler angles, phi ( $ \phi $ ), theta ($ \theta $ ), and psi ( $ \psi $ ), one for each axis. You can imagine as a plane's pitch increases positively (rotates up about the y-axis, similar to taking off), the Euler angle $ \theta $ will increase. One thing we should make sure to explore is how a plane can increase its respectively Euler angles. Let's first take a look at how a plane rolls (alters its Euler angle $ \phi $).
Fig. 2 - Airplane Roll
An airplane rolls by adjusting the position of its ailerons. The ailerons are the flaps on the back of a wing. They can rotate up and down as shown in Figure 2. The reason they cause the plane to roll is because as they rotate down, for example, the air creates a force in the negative x direction (towards the back of the plane) as well as the negative z direction (upwards). And if the aileron is rotated up, the air creates a force in the negative x direction (towards the back of the plane) as well as the positive z direction (downwards).
This creates an overall moment about the x-axis that causes the plane to roll, hence, changing $ \phi $! Now let's look at how a plane alters its pitch.
Fig. 3 - Airplane Pitch
An airplane alters its pitch by changing the position of its elevators. Elevators are on the back tail of an airplane and can rotate up and down. When they rotate up, like in Figure 3, the pitch of the plane increases (angles up), and when they rotate down, the pitch of the plane decreases (angles down). This is due to the force of the air on the elevator when it rotates. If it's rotated up, the air will create a force on the elevator in the negative x direction (towards the back of the plane) as well as the positive z direction (downwards).
This creates a net positive moment about the y-axis and, hence, increases the pitch ( $ \theta $) of the plane. Finally, let's take a look at how a plane changes yaw.
Fig. 4 - Airplane Yaw
A plane yaws by deflecting its rudder. The rudder is the structure that extends vertically from the tail and can deflect left and right. A deflection to the right, as in Figure 4, causes the plane to increase its yaw angle because the air creates a force in the negative x direction (towards the back of the plane) as well as the negative y direction (left). If the rudder were to be deflected to the left instead, the air would create a force in the negative x direction (towards the back of the plane) as well as the positive y direction (right).
This overall net moment would cause a change in the yaw angle ( $ \psi $ ).
Now that we have a better understanding of Euler angles, we should talk about their orientation. Euler angles are often read in a sequence: 3-2-1, 3-1-3, 1-2-3, etc. What we will more commonly use for this guide is 3-2-1, and what this means is there is a change in the yaw, then the pitch, then the roll. This means 3 corresponds to yaw, 2 to pitch, and 1 to roll.
So for example, a sequence of 2-3-1 would correspond to a change in the pitch, yaw, and roll, respectively. This will come in handy when we talk about rotation matrices as well!